The Price equation. (C.M. Lively for Evolution L567, Indiana University, Nov 2013)
Let’s start with selection at the level of the group. Suppose we have j groups, each with i
individuals per group. Let
be the average breeding value over all groups.
By the full Price equation, we have
,
where qj is the frequency of groups having group fitness Wj, where Wj is absolute group fitness
which is equal to
where the second “expectation” term is the mean of the products:
Now let pij equal the frequency of individuals in population j with breeding value i. As such,
is equal to:
where Wij is the absolute fitness of individual i in group j. Thus the whole puppy containing
selection within and between groups is:
Assuming no transmission bias during meiosis the Expectation term within brackets is zero.
Hence for two levels of selection, we get
where the first covariance term on the RHS is selection between groups, and the second
covariance term on the RHS is selection within groups (averaged over all groups).
If there is only one group, we get
since z is a breeding value, then var(z) is the variance in breeding values, which is the additive
genetic variance. Thus we recover the breeder’s equation (division by Wbar gives mean
relative fitness, which is what we used previously.)
Finally, if the trait is fitness, we have
which gives Fisher’s fundamental theorem of natural selection: the change in mean fitness is
equal to the additive genetic variance for fitness (divided by mean fitness).
Hamilton’s rule can also be derived from the Price equation…
Derivation of the Price Equation
But how is even one level of the price equation derived from first principals? See A. Gardner,
Current Biology vol 18 no 5. (see “GROUP SEL.pptx” for slides below). Note, the X does not just have to
out run the bear. It has to outrun all the other As that are outrunning bears in different pops.
Note: in the second to last line, the first term on the RHS is the mean of the products.
The second term on the RHS is the product of the means. The mean of the products
minus the product of the means is a covariance. The third term on the RHS is the
mean (or E for expectation) of the product of Wj and delta zj. The last line can be
rewitten to give the first equation on page 1: